A large amount of effort has been devoted to developing, enhancing and using grid generation techniques, through solution of elliptic partial differential equations (“PDEs”). Elliptic grid generation methods generally focus on developing body-conforming grids around bodies for simulations of external fluid flow. The grids thus generated are smooth, having at least continuous first and second derivatives, appropriately stretched or clustered, and are orthogonal over most of the grid domain. Inclusion of inhomogeneous terms in the PDEs allows a grid to satisfy clustering and orthogonality properties in the vicinity of specific surfaces in three dimensions and in the vicinity of specific lines in two dimensions.
Following the work of Thompson, Thames and Mastin, Jour. Computational Physics, vol. 24, 1977, pp. 274–302, three-dimensional governing equations for elliptic grid generation are often expressed as:ξxx+ξyy+ξzz=P(ξ,η,ζ)=−ai·sgn(ξ−ξi)exp{−bi|ξ−ξi|},  (1A)ηxx+ηyy+ηzz=Q(ξ,η,ζ)=−ci·sgn(η−ηi)exp{−di|η−ηi|},  (1B)ζxx+ζyy+ζzz=R(ξ,η,ζ)=−ei·sgn(ζ−ζi)exp{−fi|ζ−ζi|},  (1C)where ξ, η, and ζ are generalized curvilinear coordinates, x, y and z are Cartesian coordinates, and P(ξ,η,ζ), Q(ξ,η,ζ) and R(ξ,η,ζ), are inhomogeneous terms, ai, bi, ci, di, ei and fi are manually selected constants, and the subscript “i” refers to a particular boundary component associated with the problem.
In a two dimensional study by Steger and Sorensen, Jour. Computational Physics, vol. 33, 1979, pp 405–410, the authors use the following governing equations,ξxx+ξyy=−ai·sgn(η−ηi)exp{−di|η−ηi|},  (1D)ηxx+ηyy=−ci·sgn(η−ηi)exp{−di|η−ηi|},  (1E)for a given η boundary. The quantities ai and ci are generalized to functions ai(ξ) and ci(ξ), respectively, and the values of these quantities are computed as part of the solution by requiring a specified spacing between a given η boundary and an adjacent grid line, and grid orthogonality at this η boundary. In any two dimensional problem, the decay parameters, such as di, must be prescribed or manually inserted for each of the boundaries in any coordinate direction. However, as will be seen in the subsequent development, these values of di are coupled with the values computed for the quantities ai and ci respectively so that explicit prescriptions of values for the parameters di are in conflict with the values computed for ai and for ci. Further, the process of selecting the two values for the parameters di for two opposing boundaries, and, by extension, four values for four boundaries in a two dimensional problem, is cumbersome for static grids and is infeasible where dynamically changing grids are required. In three dimensions, the parameter values need to be prescribed for six boundaries, one for each of six boundaries.
What is needed is an approach that provides an automatic procedure for generating an elliptic grid that does not require manual insertion or user prescription of these decay parameters for a two dimensional or three dimensional grid generation problem. Preferably, these decay parameters should allow for a variable rate of decay from different points on any grid boundary, should arise automatically in the formulation and solution of the problem and should permit an interpretation in terms of one or more physical quantities associated with the problem.